Numb3rs 517: First Law

DARPA's supercomputer Bayley is the prime suspect in the murder of an
AI researcher. According to DARPA's team, Bayley represents their
success
in the creation of Artificial Intelligence and they are not accountable
for its actions. In this episode Charlie, Amita and the FBI try to
crack this case to confirm if DARPA's claims hold to be true and Bayley
has passed the Turing Test.

Alan Mathison Turing was a British mathematician, logician and computer
scientist born in London in 1912. A defender of Artificial
Intelligence, Turing played an important role in the development of
computers in Britain. Together with his friend David Champernowne he
invented "round-the-house" chess: after you move, run around the house,
if you get back before your opponent's move you are entitled to a new
move.

The Turing Test is the most known test to verify Artificial
Intelligence.
It was proposed by British mathematician Alan
Turing (see Tangent) in an article published in 1950 in
the journal "Mind" and titled
"Computer machinery and intelligence".

Below, we will briefly explain some aspects of the Turing Test and the
concept of Turing Machine. Alan
Turing introduced Turing Machines in an earlier
paper published in
1937 in the
Proceedings of the London Mathematical Society and titled "On
Computable
Numbers, with an Application to the Entscheidungsproblem". Turing's
work on computability and artificial intelligence has been
so influential
that many
consider him the father of modern computer science.

The Turing Test

Alan Turing created the Turing Test after considering the
following question: Can machines think? Due to the
controversial nature of the question he proposed an alternative
question,
namely: Are there
machines that can do well in the Turing Test? The
first
version of Turing's test, or
the "imitation game" as Turing introduced it in his paper, can be
described as
follows:

Suppose that three people take part of a game: A man (A), a women (B)
and an interrogator (C), who can be
of either sex. The interrogator (C) is in a different room than (A) and
(B). (C)'s object of the game is to identify who of the two is
the man
and
who is the women. In order to do so, (C) is allowed to address
questions
to (A) and (B). However
(C) never hears the answers directly from (A) and (B) but through a
teleprinter or intermediary. The objective of (A) is to trick (C) into
a
wrong identification, while
the objective of (B) is to help the interrogator.

Turing proceeds to ask the following question:

"What will happen when a machine takes the part of (A) in this game?"

If the interrogator decides wrongly as often when
the game is played with the computer as he does when the game is played
between a man and a woman,
it may be argued that the computer is intelligent. After this
introduction Turing gives an example of a hypothetical
conversation between the interrogator and player (A):

C:
Please write me a sonnet on
the subject of the Forth Bridge [a bridge over the Firth of Forth, in Scotland].

A: Count me out
on this one. I
never could write poetry.

C: Add 34957 to
70764.

A: (Pause about
30 seconds and
then give the answer) 105621.

C: Do you play
chess?

A: Yes.

C: I have a K
at my K1, and no
other pieces. You have only K at K6 and R at R1. It is your move. What do you
play?

A: (After a
pause of 15 seconds)
R-R8 mate.

It might be hard to notice
at first glance but after a long delay the
witness (A) makes a mistake in the arithmetic question. It might be
argued that this is a sign of human behavior. However, if the
respondent were a machine there are some possible explanations. For
instance, a hardware error or a ploy inserted by the machine's
programmer to trick the interrogator.

Activity 1:

Analyze the conversation above by giving possible explanations to the
answers if the
respondent were a machine. What questions would you ask if you were
playing the role of (C) in the "imitation game"? Why would you ask
these
questions? How could a machine trick you
with its answers?

Aware of the controversial nature of the problem, Turing lists some
objections that
could arise on the notion that machines could think. Below we will
mention a few of them:

The Theological Objection: Thinking is a function of the
soul and God reserved it for humans but not for animals or
machines.

The Mathematical
Objection: There
exist mathematical statements that can be seen to be true by humans but
cannot be mechanically proven. Hence, there does not exist a machine
that represents the human mind.

Arguments from disabilities: A
machine can mimic many human behaviors but there are numerous human
features that cannot be mechanically replicated. Among them are: Be
kind, friendly, beautiful, have a good sense of humor, etc.

Activity 2

Think of further objections you might have to the notion of artificial
intelligence and figure out possible counter-arguments to your own
objections.

For a more detailed discussion on the Turing Test we
strongly recommend Chapter XVIII of Douglas R. Hofstadter' book "Gödel,
Escher, Bach: an Eternal Golden Braid". In what follows we will explain
the concept of Turing
Machines.
In the "imitation game" described above the machine playing the role
of (A) is understood to be a Turing Machine. This type of machines is
considered by Turing in all his studies on Computability
and Artificial Intelligence.

Turing machines

In 1936, Alonzo
Church solved independently the Decision Problem by
formalizing
the concept of effective calculability thorough the notion of λ-calculus.
It was
later proved that Turing and Church's results were equivalent in what
it is now known as the
Church-Turing
Thesis.

This type of machines was defined and used for the fist time
by Alan Turing in his remarkable work "On
Computable
Numbers, with an Application to the Entscheidungsproblem".
In this paper, Turing resolved a question known as
the Entscheidungsproblem
or Decision Problem. The Decision Problem conjectured the
existence of a mechanical
process to decide whether a given mathematical statement
is true or false. This problem is part of a three-fold
question proposed by David
Hilbert in 1928: Is
mathematics complete, consistent and decidable? Czech
mathematician Kurt
Gödel gave a negative answer to the first two questions with
his famous incompleteness
theorems.
By formalizing the concept of computability and algorithm
through
the concept of Turing
Machine, Alan Turing proved in 1936 the
"undecidability" of mathematics to give a negative answer to the third
part as well (see Tangent).

Informally, a Turing Machine is a theoretical device
consisting of an infinite line of cells called a tape
and an active element called head.
The head moves along the tape and has an internal state
that changes through the computation. The number of possible states of
the head is finite and there is a subset of states known as final states,
where the computation ends or halts. Each cell on the tape contains a
symbol
from a finite alphabet,
which can be read and changed by the head. There is a special symbol, B,
which is the only one that can appear infinitely many times on the tape
throughout the computation. The Turing machine has a table of
orders that specifies the new state of the head, the
symbol
written on the cell and the direction of the head's movement (left, L,
or right, R)
depending on the current state of the head and the symbol read on the
tape's cell. As the reader might notice the definition of
Turing machine is obscure and is not suitable for implementation of
algorithms. Its use is purely theoretical, rather than computational
per se. Turing machines are abstract devices used to simulate
the logic of any computer algorithm.
To clarify the definition stated above we present an example.

Example

Suppose that a Turing machine has the alphabet {0,1,B} and the
states {0,1}. The end state is 1. The table of orders is

Current state

Scanned symbol

Print symbol

New State

Move

0

0

1

0

R

0

1

1

0

R

0

B

B

1

R

Assume that the initial state is 0 and that initially the head
is
pointing at asymbol different than B
on the tape. The image below shows the partial steps in the
computation when the initial configuration of the tape is
.....BBB01BBB......

In general, for any initial 0-1 vector this machine changes
all the zeros on the tape by
ones. This simple operation needs a relatively large
description in the language of Turing machines, which highlights the
fact
that these machines are not suitable for implementation. Clearly, this
machine stops running given any initial configuration
of the tape. In
general, the problem of determining whether, given an initial
input, a Turing machine will
finish running or run forever is known as the halting
problem. Turing proved in 1936 that it is
impossible to find an algorithm to solve the halting problem. When
translated to the language of number theory this result shows the
"undecidability" of number theory and answers the Entscheidungsproblem.

Activity 3

a) Describe a Turing machine that converts all the 0's on a 0-1 vector
into 1's and all the 1's into 0.
b) Describe a Turing machine for which the computation on a
particular initial input runs forever.